where \(-\lambda \)  is the inverse of \(\lambda \) . Then \(P_\mathfrak {G}(\lambda )\)  is a closed \(k\) -subgroup of \(\mathfrak {G}\)  ([1]}), \(U_\mathfrak {G}(\lambda )\)  is an affine algebraic \(k\) -normal subgroup of \(P_\mathfrak {G}(\lambda )\) , and \(Z_\mathfrak {G}(\lambda )\)  is the centralizer of the \({G}_m\) -action in \(\mathfrak {G}\)  ([1]}). By [1]}, these subgroups are smooth because \(\mathfrak {G}\)  is smooth.
